The level sets of a cubic polynomial in four or more variables tend to have many integer solutions, while ones in two variables a limited number of solutions. Very little is known in the case of three variables. For cubics which are character varieties (thus carrying a non-linear group of morphisms) a Diophantine analysis has been developed and we will describe it. Passing from solutions in integers to integers in, say, a real quadratic field there is a fundamental change which is closely connected to challenging questions about one-commutators in 𝔰𝔩2 over such rings.
This is the second of a two-part lecture, the first of which is here.
This video is part of Harvard University‘s two-part Alfors lecture 2019.
